exercise:Ef5bcf05aa: Difference between revisions
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(Created page with "<div class="d-none"><math> \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}</math></div> Given a random walk <math>W</math> of length <math>m</math>, with summands <math display="block"> \{X_1, X_2, \ldots,X_m\}\ , </math> define the ''reversed'' random walk to be the walk <math>W^*</math> with summands <math display="block"> \{...") |
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<ul><li> Show that the <math>k</math>th partial sum <math>S^*_k</math> satisfies the equation | <ul style="list-style-type:lower-alpha"><li> Show that the <math>k</math>th partial sum <math>S^*_k</math> satisfies the equation | ||
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<li> Use parts (a) and (b) to prove [[guide:A6f1984402#thm 12.3.2 |Theorem | <li> Use parts (a) and (b) to prove [[guide:A6f1984402#thm 12.3.2 |Theorem]]. | ||
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Latest revision as of 01:08, 15 June 2024
[math]
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Given a random walk [math]W[/math] of length [math]m[/math], with summands
[[math]] \{X_1, X_2, \ldots,X_m\}\ , [[/math]]
define the reversed random walk to be the walk [math]W^*[/math] with summands
[[math]] \{X_m, X_{m-1}, \ldots, X_1\}\ . [[/math]]
- Show that the [math]k[/math]th partial sum [math]S^*_k[/math] satisfies the equation
[[math]] S^*_k = S_m - S_{n-k}\ , [[/math]]where [math]S_k[/math] is the [math]k[/math]th partial sum for the random walk [math]W[/math].
- Explain the geometric relationship between the graphs of a random walk and its reversal. (It is not in general true that one graph is obtained from the other by reflecting in a vertical line.)
- Use parts (a) and (b) to prove Theorem.